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1answer
26 views

Tail bounds of reciprocals

Suppose one knows that for a random function $f(n)$, $f(n)-a$ decays at some rate given by: $$Pr(|f(n)-t|>\epsilon)=g(\epsilon),$$for $g(\epsilon)\to0$, all as $n\to\infty$. If the above holds, ...
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1answer
30 views

Scaling Cumulative Probability Distribution function values

We have a cumulative probability distribution function (cdf), we want to scale it down for using it in anomaly detection. The mapping should look like this. CDF value: 0.1 ... 0.5 ... 0.9 ... ...
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20 views

A more general class for power-law distribution

Is a power-law distribution, also a: Fat-tailed distribution? Heavy-tailed distribution? Long-tailed distribution? Also, which of the three distributions above is a subclass of the other?
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31 views

Uniform bound of linear function of sub-Gaussian random variable over a compact set

Problem: Let $K\subset \mathbb{R}^p$ be a compact set that admits an $\epsilon$-net $\mathcal{N}_\epsilon$ with respect to the Euclidean distance of $\mathbb{R}^p$, and $|\mathcal{N}_\epsilon|\leq ...
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0answers
18 views

Truncated distributions with out of bounds values set equal to the value of truncation

Suppose we have a set of integers which are not uniformly distributed over the range {0 ... 100} (some gamma distribution) Suppose then that we transform this set such that all integers greater than ...
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1answer
31 views

Tails of a random variable

I would like to know if there are any standard tests or approaches to checking whether the following holds: $$ \limsup_{x \rightarrow \infty} x \mathbb{P}(X \geq x) < \infty $$ where $X$ is a ...
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1answer
29 views

Question about relating variance to existing tailbound. Apparently “straightforward”.

I'm trying to make sense of a claim made in this paper (in particular justifying the equation in between equations 25 and 26). Suppose $X$ is a random variable with mean zero, and we know that ...
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0answers
56 views

Lower bound for upper tail of binomial distribution

I need a lower bound for the upper tail of $X \sim Bin(n,\beta^m)$. In general concentration inequalities (Hoeffding) give us upper bounds for $\mathbb{P}(X>t)$ (if $t > \mathbb{E}[X]$), but no ...
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0answers
21 views

Looking for a certain probability distribution

I would like to know if one can find a probability distribution with finite mean and the following property: $F(2^{i+1})-F(2^i)\le p$ for given parameter $0<p<1$ all $i$. That is, if we ...
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1answer
43 views

Sum of slowly varying functions

We call a function $L:(0,\infty)\rightarrow(0,\infty)$ slowly varying if for each $c>0$ one has $\displaystyle \lim_{x\to\infty}\frac{L(cx)}{L(x)}=1$. Can somebody give me a hint why the sum of ...
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1answer
62 views

Multinomial distribution: bounding sum of coordinates' deviations from mean

Okay, given a multinomial random vector $$X=\text{Multinom}(n,\;p_{1},\;\dots,\;p_{k}),$$ so that $$X=(X_{1},\;\dots,\;X_{k})\;\;\;\text{with} \;\;\;\sum_{i=1}^{k}X_{i}=n,$$ I'm looking for a bound ...
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0answers
28 views

Chernoff Bounds for non-Bernoulli random variables

Normally, one applies the Chernoff bound for a sum of independent Bernoulli random variable. But what about if there are other outcomes? For a sum of random variables with outcomes $0, c$, one can ...
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0answers
112 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
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2answers
56 views

Variation of Chebsyhev: How to prove that?

I have the "job" to prove that for any random variable with standard deviation $\sigma$ and expectation $\mu$ and for any $t>0$ we have $$Pr[X-\mu \geq t \sigma] \leq \frac{1}{1+t^2}.$$ I thought ...
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1answer
46 views

Tail estimates for Binomial with constant mean

The Chernoff Bound gives a good tail estimate for a Binomial Distribution, but only if the mean goes to infinity. However, for a constant mean, Chernoff bound does not help at all. Is there some ...
4
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1answer
101 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
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1answer
81 views

How to Prove this Multinomial Distribution Inequality

I have the following lemma, but there seems to be one (or two) mistakes in the proof found in this paper (lemma 3). The lemma states that for $Multinomial(n,p_1,\ldots,p_k)$ distributed ...
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0answers
214 views

Concentration inequality of weighted sum of random variables given a tail inequality

I tried to solve the exercise below which can be seen as a generalization of the Bernstein's concentration inequality. However, I have difficulty bounding the moment generating function of $Z$ (see ...
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1answer
41 views

limiting behavior of standard normal survivor function [duplicate]

How do you show that $\lim_{x\to \infty} 1-\Phi(x) \sim \phi(x)/x$? In the previous, I'm using $\Phi$ to refer to the standard normal CDF and $\phi$ to refer to the standard normal pdf. Thanks!!
0
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1answer
160 views

Levy flight distribution

Can somebody help me with C++ code, how to make a Levy distribution like a function? I need to make one dimensional Levy flight model, but I don't know the function how to choose the right step.
2
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1answer
112 views

Is Wikipedia correct?

Cantelli's Theorem Wikipedia says: $$P[X-\mu \geq a] \leq \frac{\sigma^2}{\sigma^2+a^2}$$ for $a > 0$ and $$P[X-\mu\geq a] \geq 1- \frac{\sigma^2}{\sigma^2+a^2}$$ for $a <0$. Is the second ...
1
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1answer
38 views

“Reverse” distribution-tails

Chernoff, Markov and Chebyhev all give some upper bound for tail probabilities, e.g. Chebyshev gives us $Pr[|X-E[X]| \geq t] \leq \frac{Var[X]}{t^2}$. This is quite helpful, but what if I would ...
0
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1answer
49 views

Tail of a sequence of RV

Consider the sequence of random variables $\{X_n\}_{n=1,2,\dotsc}$ where $X_n$ is gamma-distributed with shape $n$ and scale $1/n$ (or equivalently, $2nX_n$ is $\chi^2$-distributed with $2n$ degrees ...
2
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1answer
90 views

Inequalities for the tail of the normal distribution (Halfin-Whitt paper)

I am reading the famous paper by Halfin and Whitt, [1]. I'd like to prove remark (1) on page 575. The authors state \begin{align} \frac{\beta \alpha}{(1-\alpha)} = \frac{\phi(\beta)}{\Phi(\beta)} ...
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1answer
194 views

Tail bound sum of unbounded RVs of given mean and variance?

Let $X_1,\dots,X_n$ be independent variables, $X_i$ having mean $\mu_i$ and variance $\sigma_i^2$. Let their sum $S = \sum_{i=1}^n X_i$. Of course, $S$ has mean $\lambda = \sum_{i=1}^n \mu_i$ and ...
2
votes
1answer
100 views

Mnemonic for platykurtic and leptokurtic

I keep confusing terms leptokurtic and platykurtic. Is there a good mnemonic to help remember which is which? "Lepto" means "little", "platy" means "flat", and both are equally unrelated to thickness ...
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0answers
143 views

Long tail distributions

I have just started to study sub-exponential distributions. I wonder how to prove this statement : Let $X_1,\dots,X_n$ are independent random variables with distributions $F_1,\dots,F_n$ respectively. ...
1
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1answer
410 views

difference between upper and lower tails of Chernoff bounds

I'm struggling with the intuition behind why Chernoff bounds differ in the upper and lower tails. That is, for the lower tail we have: $$ Pr(X \le (1 - \delta)\mu)\ \ \le\ \ ...
2
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1answer
81 views

Conditions on r.v. $X$ s.t. $\Pr(X\ge n\mid X\ge n/2)$ gets small?

So I have a random variable $X$ that takes values in the non-negative integers. I want it to have one of the following properties: either $$\lim_{n\to\infty} \Pr(X\ge n\mid X\ge n/2)=0$$ or there ...
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3answers
362 views

Proof that $E(X)<\infty$ entails $\lim_{n\to\infty}n\Pr(X\ge n) = 0$?

As the title says. I think this should follow straightforwardly but I can't find a proof. My random variable of interest $X$ takes values in the non-negative integers. The only other assumption on ...
1
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1answer
57 views

What is the null hypothesis for a 1-tailed test?

I've been given different answers to this question in different courses. Some professors say it is (using the example alternative hypothesis of $\mu > 3$): $$H_0: \mu = 3$$ $$H_1: \mu > 3$$ ...
2
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1answer
189 views

'Simple" Chernoff bounds

Reading an academic paper I've observed the next claim: A simple Chernoff argument will now show that if an event has a constant probability at every step of occurring and there's independence ...
1
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1answer
142 views

Tail bound for hypergeometric distribution

I am looking for a reference (book) for the tail bound for the Hypergeometric distribution. I know there is a nice paper by Skala (2009) but its unpublished. I am looking for a book which would be a ...
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2answers
201 views

Coins and probability

Bob has $n$ coins, each of which falls heads with the probability $p$. In the first round Bob tosses all coins, in the second round Bob tosses only those coins which fell heads in the first round. Let ...
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1answer
236 views

Chernoff bound for Geometric RVs compared to exact tail bound

I keep getting a result I can't interpret. X is a Geometric RV with distribution ($0<\rho<1$) $$ \pi_k = \rho^k(1- \rho) $$ so directly applying Geometric series the tail bound is $$ B_1 = ...
1
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1answer
391 views

Fat Tail / Large Kurtosis Discrete Distributions?

All, I'm wondering if there are any notable, basic discrete probability distributions with "fat/heavy tails" or a large kurtosis? I know the Geometric Distribution's excess kurtosis approaches 6, ...
0
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1answer
306 views

Upper-tail inequality for t-distribution

I am interested in upper tail bounds (or bounds on deviation from the mean) for t-distribution with n degrees of freedom (http://en.wikipedia.org/wiki/Student's_t-distribution) A bound that is of the ...
6
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2answers
494 views

Repeatedly rolling a die and the tails of the multinomial distribution.

For $1\leq i\leq n$ let $X_i$ be independent random variables, and let each $X_i$ be the uniform distribution on the set ${0,1,2,\dots,m}$ so that $X_i$ is like an $m+1$ sided die. Let ...
1
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2answers
204 views

Deducing Markov inequality from reverse Markov inequality?

Let $x$ be random variable, such that $E(x)=0,E(x^2)=1$ and $P(x^2\geq s^2)\geq\displaystyle\frac{C}{s^t}$, where $C>0,s\geq 1 , t>0$. Let $m<n$ and $m,n$ are natural numbers very big. Let ...